Here's the script that I wrote:

#!/usr/bin/perl

use strict;
use warnings;

use Data::Dumper;

#  Build all possible strings based on the following three rules:
#  1. Start with 'foobar'
#  2. Optionally include first set:
#     a. zero or one 'a';
#     b. zero or one 'b'; and
#     c. zero or one 'c'.
#  3. Optionally include second set:
#     a. 'X'
#     b. zero or one 'a';
#     c. zero or one 'b'; and
#     d. zero or one 'c'.

my ( @w, %solutions );

{
  @w = ('foobar');
  for my $firstSet ( 0 .. 1 ) {
    for my $a1 ( 0 .. 1 ) {
      for my $b1 ( 0 .. 1 ) {
        for my $c1 ( 0 .. 1 ) {
          for my $secondSet ( 0 .. 1 ) {
            for my $a2 ( 0 .. 1 ) {
              for my $b2 ( 0 .. 1 ) {
                for my $c2 ( 0 .. 1 ) {
                  if ($firstSet) {
                    if ($a1) { addThisChar('a'); }
                    if ($b1) { addThisChar('b'); }
                    if ($c1) { addThisChar('c'); }
                  }
                  if ($secondSet) {
                    addThisChar('X');
                    if ($a2) { addThisChar('a'); }
                    if ($b2) { addThisChar('b'); }
                    if ($c2) { addThisChar('c'); }
                  }
                  saveThatPattern();
                }
              }
            }
          }
        }
      }
    }

  }
  for my $key (
    sort { $solutions{$b} <=> $solutions{$a} }
    sort keys %solutions
    )
  {
    print "  $key => $solutions{ $key }\n";
  }
}

sub addThisChar { push( @w, shift ); }
sub saveThatPattern { $solutions{ join( '', @w ) }++; @w = ('foobar');
+ }
[download]

When I run this, I get the following result:

tab@music3:/../Perlmongers$ perl -w 1056774-2.pl
  foobar => 72
  foobarX => 9
  foobarXa => 9
  foobarXab => 9
  foobarXabc => 9
  foobarXac => 9
  foobarXb => 9
  foobarXbc => 9
  foobarXc => 9
  foobara => 8
  foobarab => 8
  foobarabc => 8
  foobarac => 8
  foobarb => 8
  foobarbc => 8
  foobarc => 8
  foobaraX => 1
  foobaraXa => 1
  foobaraXab => 1
  foobaraXabc => 1
  foobaraXac => 1
  foobaraXb => 1
  foobaraXbc => 1
  foobaraXc => 1
  foobarabX => 1
  foobarabXa => 1
  foobarabXab => 1
  foobarabXabc => 1
  foobarabXac => 1
  foobarabXb => 1
  foobarabXbc => 1
  foobarabXc => 1
  foobarabcX => 1
  foobarabcXa => 1
  foobarabcXab => 1
  foobarabcXabc => 1
  foobarabcXac => 1
  foobarabcXb => 1
  foobarabcXbc => 1
  foobarabcXc => 1
  foobaracX => 1
  foobaracXa => 1
  foobaracXab => 1
  foobaracXabc => 1
  foobaracXac => 1
  foobaracXb => 1
  foobaracXbc => 1
  foobaracXc => 1
  foobarbX => 1
  foobarbXa => 1
  foobarbXab => 1
  foobarbXabc => 1
  foobarbXac => 1
  foobarbXb => 1
  foobarbXbc => 1
  foobarbXc => 1
  foobarbcX => 1
  foobarbcXa => 1
  foobarbcXab => 1
  foobarbcXabc => 1
  foobarbcXac => 1
  foobarbcXb => 1
  foobarbcXbc => 1
  foobarbcXc => 1
  foobarcX => 1
  foobarcXa => 1
  foobarcXab => 1
  foobarcXabc => 1
  foobarcXac => 1
  foobarcXb => 1
  foobarcXbc => 1
  foobarcXc => 1
tab@music3:/../Perlmongers$
[download]

And sure enough, foobarabcX is there. So I'm now convinced that the original specification was broken.